Mathematical device



Jam. 13, 1942. w. G. Hu'rToN MATHEMATIAL DEVICE Filed April 3Q. 1941 3 Sheets-Sheet l ,E Fja l .P Fmi s2 s, N42 f2 NLT J3 INVEN'x-on mm @Hmm WM* W Janf 13, 1942. wl G. Hun-0N MATHEMATICAL DEVIQE 3 Sheets-Sheet 2 Filed April 30, 1941 swr new..

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Jan. 13, 1942. w, G, H'UTTON 2,269,915

MATHEMATICAL DEVICE Filed April 30, 1941 3 Sheets-Sheet 3 INVENTOR.

He Y M/ffamwmf Q/PMM Wm sign an array ot antennas to give a dennite de- Patented Jan. .13, 1942 e UNITED- STATES PATENT lOFFICE MATHEMATICAL DEVICE William G. Hutton, Parma, Ohio, assignor oi' one-third to Windsor H. Atwater, South Euclid, Ohio, and one-third to Robert Morris Pierce, Lakewood, Ohio Application April A30, 1941, Serial No. 391,157

20 Claims. (01.'235-61) My invention relates in general to a mechanism In the design of an antenna array, it is elecfor solving equations which may be solved vectrically possible to mould the radiation pattern torially and more particularly to a mechanical in almost .any desired manner. As the number directional antenna pattern calculator. of radiators or antennas is increased, a greater While I will describe my invention as being em- 5 degree of control becomes possible. The descripbodied in a mechanical directional antenna pattion will deal with a three-element array, but tern calculator, yet it is to be understood that it is to be understood that my invention may be my invention may be embodied in other forms applied to any number of 1 elements. For a to solve other problems of a similar or varied three-element antenna array, the following nature. l, equation is accepted as a standard for calculat- The outstanding problem in antenna design is ing the relative iield intensity in any direction, always to control the distribution of the radiated both horizontally and vertically. The relative energy in some desired manner. The radiation iield intensity, represented by e, is:

pattern, or space characteristic, is a geometrical When the antenna heights are equal, Equation description of the manner in which the radiant 1 slmpliiies to:

energy distributed in space around the radi- The nomenclature used is:

ators or antennas. The horizontal pattern rep- N I resents the distribution in the ground plane, and gg: gif :':tieeelqoe'z the nous Vertical plane patterns represent 30 la=height of antenna element No. I, degrees. conditions in directions at various angles above v mzbearmg of une drawn from antenna Nm I' the horizontal. The patterns are represented in which s used as reference point to am polar graphs andmay be drawn in terms of rela- A tenna N0 tive or actual ileld strength or power. The present ,description will be in terms of relative field eezbeermg er eee drawn teem entenne No' I mtens1t1e5 35 to antenna No. I.

'rne neldvintensity in any direction from one '#:azmuth angle (um degrees is wundere or more antennas which radiate power at the reference direetiem' in tenu N. and same frequency can be calculated from a standse epee e degrees between en e e I antenna No. 2. ard equation, provided sumcient conditions are known conce g the antennas their ump 40 S: spacing, degrees, between antenna No. and

f antenna No. I. ment and the distribution of'power to them. l The. p of antennas to radiate power u. 0 a ngle (vertica is considered zero de spoken oi as the antenna array. The greater :um hm 1 t cment'nowm m the number of antennas in an array, the more e, e p ene e e e time and work is required to 'calculate the radiation pattern ior that array. In order to deing in antenna No. I. l fs=time phase angle o1' current ilowing in antenna No. I with respect'to current nowsired pattern, it is often necessary to determine, m8 in antenna No. L

by trial. the radiation pattern for several arrays 50 v1, vl, vi=the magnitude' of the eir'etlve nelds before the correct may is oe1eotod Cowradiated by the antenne elements Nos. l,

quently, a great amount of time and effort is 2 and 3, respectively, and are proportional entailerlinizhedesignoilanantennaarray iorthe to the currentsiiowlng in the/respective Purpose of getting a desired ileld intensity patv antenna elements. tern. Il lc=arbitrary constant.

tenna No. I with respect to current iiow- An object of my invention is the provision of a mechanism adapted to solve equations of the type which may be solved vectorially.

Another object of my invention is the provision of a mechanism adapted to solve equations which may be represented by the general vector equation, e:k(V1+V2+ -i-Vn), hereinafter referred to.

Another object of my invention is the provision of a mechanism which will solve for the value of the radical expressed in the Equation 1 and 2 above.

Another object of my invention is the provision of a mechanism for generating a movement corresponding to a cosine function and for transferring the said movement to the rotation of vectors so related to each other and to a fixed vector that the vectorsum may be read directly at any position throughout the complete cycle of the generated movement.

Another object of my invention is the provision of a mechanism for generating a movement corresponding to a cosine function and for transferring the said movement to a plurality of conditions which when affected by the generated movement provideifor giving a resultant condition a't any position throughout the complete cycle of the generated movement.

Another object of my invention is the provision of a mechanism for generating recurrent move'- ment and for transferring the said movement to a plurality of conditions which when affected by the generated movement provide for giving a resultant condition at any position throughout the complete cycle of the generated movement.

Another object of my invention is the provision of varying the generating recurrent motion as to amplitude and phase.

Another object of my invention is the provision of adjusting thegmagnitude and angle of a vector with respect to a reference position.

Another object of my invention is the provision of a mechanism adapted to facilitate the calculation of the relative field intensity of two or more antenna elements in, any direction, both horizontally and vertically.

Other objects and a fuller understanding of my invention may be had by referring to the following description and claims, taken in conjunction with the accompanying drawings, in which:

Figure 1 is a representation in a horizontal plan view of`the spacing and the orientation of a three-element antenna array, as set oil' on my calculating machine for solving for one value of the eld in the horizontal pattern, where angle :zero;

Figure 2 is a representation of the position of the vectors of my calculating machine, as obtainable by the setting in Figure 1 for solving for the said one value of the field in the horizontal pattern; Y

Figure 3 is a representation similar to Figure 1 for solving for another value of the field in the horizontal pattern with the three-element antenna array rotated through an angle o! :'10 degrees Figure 4 is a representation of the position of the vectors of my calculating machine, as obtainable by the setting in Figure 3 for solving for the said another value of the field in the horiaontal pattern;

Figure 5 is a polar diagram of a horizontal radiation pattern of the three-element antenna array. as obtainable by my calculating machine,

the resultant relative field for the settings in Figures l and 3 being shown at angles of zero and 70 degrees respectively;

Figure 6 is a plan view of the three-element antenna array as actually installed in service;

Figure 7 is a plan view of a. calculating machine embodying the features of my invention and is adapted to facilitate the calculation of the relative eld intensity in any direction, both horizontally and vertically; i

- Figure 8 is a. representation similar to Figure 3 with the three-element antenna, array rotated through an angle of :20 degrees, being a representation corresponding to a setting on my machine utilized in determining the setting of my machine to solve for values representing a function of the field in the vertical pattern;

Figure 9 is a representation based upon the setting in Figure 8 and is an elevational view for :20 degrees and 0:60 degrees and serves to illustrate how my machine may be set ofi to solve for values representing a function oi the field in the vertical pattern;

Figure 10 is a representation of the position of, vectors of my calculating machine, as obtainable for the setting in Figure 9, :20 degrees and 0:60 degrees, for solving for a value representing a function of the iield in the vertical pattern;

Figure 1i is a vertical pattern at :20 degrees, when all antennas are 90 degrees in height;

Figure 12 is a representation similar to Figure 9 and is an elevational view for 95:70 degrees and 0:40 degrees, and illustrates how my machine may be set off to solve for values representing a function of the field in the vertical pattern;

Figure 13 is a representation of the position of the vectors of my calculating machine, as obtainable for the setting in Figure 12, :70 degrees and 0:40 degrees, for solving for a value representing a function of the field in the vertical pattern; and

Figure 14 is a vertical pattern at :'70 degrees when all antennas are 90 degrees in height,

'Ihe Equations l. and 2 above for 'a three-element array is derived from the vector addition of the three vectors which represent the strength of the signal from each of the three antennas arriving Yat a common point far removed from the antenna array. Each of the three vectors ls dependent only on the radiation from its' respective antenna with respect to the reference point or origin. A study of these vectors and their relationship with each other will now be developed, in conjunction with Figure 1 of the drawings, which diagrammatically illustrates a threeelement array in horizontal plan view, as set off on my machine in Figure 7.

Antenna No. I is chosen as the reference point or origin O, and the angle :zero degrees, establishing a directional reference line for antenna No. I. Antenna No. 2 is spaced S2 degrees from antenna No. I in a direction pz degrees from antenna No. I or .from the reference line':0, drawn through No. il.) Antenna No. 3 is spaced S: degrees from antenna No. I at an angle of o: degrees from the reference line for antenna No. I. The current in antenna No. I is assumed to have a phase of zero degrees, while that of antenna No. 2 is lf2 and that of antenna No. I is #a degrees with respect to antenna No. I. The point of observation P may be considered to be anywhere on a circle drawn from the origin O, and it is assumed that the distance from the is an integral number of Wave-lengths from O, l

which means that the signal from antenna No. I is in time phase with the current in antenna No. I.

Thus, the vector representing the signal from'antenna No. I being a reference vector, may be drawn at an angle of zero degrees and vits length will depend upon the strength of the signal from antenna No. I. Antenna No. 2 is farther away lfrom-P than antenna No. I by a magnitude, Sz.,

cos (o2-1p) degrees. As a result, the signal from antenna No. 2 will arrive at P behind the signal from antenna No'. I due to the longer path. The delayed arrival of the signal from antenna No. 2

to P due to the longer path is compensated forv or overcome by reason of the fact that the sig- 1 nal from antenna No. 2 starts with phase 1pz ahead of signal from antenna No. I. The combination of space/and phase results 'in an equivalent phase P by an equivalent phase of, S3 cos (4m-qb) +1,03

l written:

where Y ifi represents vector No. I, and includes both magnitude and angle,

-z represents vector No. 2, and includes both magnitude and angle, and

n represents vector No. n, andincludes both magnitude and angle.

For a complete picture of the field pattern for an array, such forexampleas the pattern shown f in Figure 5, the total eld `rlnust be found for P, a number of times and where P is located at a. different angle for the array each time. Usually the field is calculated for P located at intervals of 5 or 10 degrees or any other value throughout the horizontal plane. P in Figure l, being a point at which the eld intensity is determined, is but one of several points located radially about the origin O for the three antennas Nos. I, 2 and 3. In the actual field layout, the antennas remain fixed and the observation point is moved in a clockwise direction aboutthe origin of the'array.

The embodiment ofv my invention as shown in Figure 7 is designed to determine the field pattern for a three-elementarray. It 4can likewise be used directlyfor a two-element array. The principle may be also extended to apply to more than three elements. Although the device operates to solve radiation problems, its use is not limited to their solution only. It may be easily adapted to the solution of problems not related to radiation, which involve functions equivalent to those found in the equation for field intensity for antenna arrays or to those which may be solved vectorially. With particular reference to Figure 7, my mechanicaldik [cos l-cos' (l cos 6)] e# sin 0 Since sin 6=sin 90 degrees=1, (vertical isvconsidered zero degrees, as 0=zenith angle), the Equation 3 may -be expressed as follows:

+(V, sin .12+ V3 sin a, 2 www The solving of the radical in Equation 4, being a function of the total field intensity at P due to the radiation from the -three-element antennas may be determined as shown inV Figure 2 by vector addition, where: .i

By assuming that the value of the radical is the resultant of a sum of group of vectors, then Equation 4 may be Written vectorially as:

rectional antenna pattern calculator comprises generally three main parts; namely, a cosine generator indicated generallyby the reference character |52, a vector system indicated generally by the reference character I6, and motion transi mitting means indicated generally by the reference character I1 for transmitting the motions of the cosine generator I5 to the vector system I6. The cosine generator I5 generates a movement corresponding to a cosine function, and the motion transmitting means I1 transmits vthis motion to the rotation of vectors in my vector system, wherein the vectors are so related to each other and to a fixed vector that the vector sum may be read directly at any position throughout the complete cycle of the generated. movement. The cosine generator I5, the vector f system I6, and the motion transmitting means may comprise a fiat'sheet of material of thel e--KV (5) where V represents the resultant vector which is equal in magnitude to the radical value.

The vector Equation 5 may be written in its vector component form asi follows:

.an adjusting arm 20 'mounted on top of the l plate 24 and the antenna A3 is carried by -an e=K v1+v2+ +ve (6) 75 I'I are suitably mounted upon a base 24 which proper dimensions and thickness.

In th'ecosine generator arrangement, the ref-y erence character A1 represents antenna No. I, the reference character A2 represents antenna No. 2, and the reference character A3 represents antennaNo. 3. The antenna A1 is located at the origin, whereas the antenna A2 is carried by adjusting arm 29 mounted upon the underneath side of the plate 24'. The antenna Az and the antenna Ag are arranged to be revolved about That is to say, the point the origin or the antenna A1 by means of a pointer indicated generally by the reference character I. The pointer I may be rotated throughout a complete circle of 360 degrees which is marked off on the base plate 24 at intervals of degrees. vThe angle between the pointer I and the horizontal passing through the zero mark represents the angle o and the direction of the pointer I represents the direction toward the observation point P. The adjusting arm which carries the antenna A2 is arranged to be adjustably mounted upon an elongated block 2i which in turn is carried by a small circular plate 22 arranged to be rotated by the pointer I. The small circular plate 22 is positioned on a large circular plate 23 and the arrangement is such that the small circular plate 22 which carries the adjusting arm 2B may be rotated with reference to the large circular plate 23 by loosening the screws 25 and then turning same to adjustably position the angular location of the antenna A2. When the angular position of the antenna A2 is once located, the screws 25 may again be tightened for anchoring the small cir.

cular plate 22 to the large circular plate 23. The pointer I is directly connected to the large circular plate 23 and movements thereof are then transmitted to the small circular` plate 22, the elongated block 2| and the adjusting arm 20 to the antenna A2. The adjusting arm 20 may be adjusted radially with reference to the origin O or the antenna A1 by loosening the clamp screw 21 which permits the adjustable arm 20 to be moved longitudinally Within the slot of the elongated block 2l which is anchored to the small circular plate 22 by means of the screws 26. When the adjusting arm 20 is properly set, the clamping screw 21 may be tightened for securing the adjusting arm 20 in a xed position. A graduated scale 28 is provided upon the adjusting arm 20 to indicate the spacing between the antenna A2 and the antenna A1 at the origin. The adjusting arm 29 which carries the antenna A: and which is mounted upon the underneath side of the plate 24 is arranged to be adjustably positioned in the same manner as that described with reference to the adjusting arm 20 and the antenna A2. That is to say, the antenna Aa may be angularly adjusted with respect to the pointer I and may likewise be radially adjusted with reference to the spacing from the antenna A1 at the origin. The construction for the mounting of the adjusting arm 29 on the underneath side of the plate 24 is substantially the same as that described with reference to the mounting of the adjusting arm 20 mounted on top of the plate 24.l The setting for the antennas Az and As is the same as that shown diagrammatically in Figure 1 of the drawings, wherein the angle a for the antenna A2 is 210 degrees and the angle 3 for the antenna A: is 330 degrees. In my cosine generator l5 shown in Figure '1, the angles 2 and o: are measured in a counter-clockwise direction whereas in Figure 6 which represents the actual field location of the antennas the angles p2 and 3 are measured in a clockwise direction. The reason for this is that in the field the observer at the point P is considered as moving bodily in a circle around the origin of the antennas whereas in my machine the observer is considered as remaining stationary and the antennas A2 and A: are rotated bodily about the antenna A1 at the origin. Thus, in setting off the angular position l of the antennas A2 and A3 for the angles pz and azeojovs a, the inner circular scale is used. The outer circular scale is used for the pointer I to indicate the angle p between the pointer I and the horizontal line passing through 0 degrees. The end of the adjusting arm 20 which carries the antenna A2 is arranged to slide in an elongated slot within a slotted arm 33 having its upper end securely fastened to a traverse member 34 of my motion transmitting means I1. Ihe traverse member 34 is arranged to be longitudinally movable between two spaced fixed rollers 35 on one side and two spaced pivotally mounted rollers 36 upon the other side. The pivotally mounted rollers 36 are carried by a relatively short arm which swings about a pivot point 38 and is adapted to urge the rollers 36 against the side of the traverse member 34 under the tension of a spring 39. The arrangement of the rollers is such that the traverse member 34 may be freely moved longitudinally as the antenna A2 is revolved by theactuation of the pointer I. In other words, the movements of the traverse member 34 correspond to a cosine function of the angular movements of the pointer I with reference to the horizontal reference line passing through the zero degrees. By a similar construction as that just described, the endof the arm 29 which carries the antenna A3 upon the underneath side of the plate 24 is arranged to slide within an elongated slot in a slotted arm 4D having its upper end securely connected to a traverse bar 31 mounted upon the underneath side of the plate 24. The traverse member 31 is likewise adapted to move longitudinally between rollers mounted upon the underneath side of the plate 24 in the same fashion as therollers 35 and 3B are mounted upon the top side of the plate 24. Accordingly, the longitudinal movements of the traverse member 31 correspond to a cosine function of the angle between the pointer I and the horizontal line passing through the zero degree mark. The longitudinal movements of the traverse member 34 and the traverse member 31 are transmitted to my vector system indicated generally by the reference character I6.

In my vector system, the points w, y and z represent corners of a vector figure such, for example, as shown in Figure 2 of the drawings. In my machine, the point w is the center of a rotating shaft 42, rotatably mounted in the base plate 24, and carries a pulley 43 positioned upon the underneath side of the plate 24. The pulley 43 is driven or rotated by a flexible member 44 such, for example, as a flexible wire. One end of the wire is anchored to an anchoring block 46 carried upon the right-hand of the traverse member 31 and the other end of the wire is anchored to a spring 41 which is fastened to an intermediate portion of the traverse member 31. The wire upon leaving the anchoring block 4B passes over a pulley 45, upon the left-hand side thereof, then across to the pulley 43 where it is Wrapped around one or more times, after which the wire extends across to the right-hand side of the pulley 45 and thence tothe spring 41 which is fastened to an intermediate portion of the traverse member 31. The spring 41 maintains the wire under relatively rigid tension so that the longitudinal movements of the traverse member 31 are transmitted to the pulley 43 for rotating the shaft 42. To the upper end of the rotating shaft 42 on the upper side of the plate 24, is mounted a circular graduated scale member 68 and an elongated block 48 in which is adjustably mounted an adjustable vector arm `reference character 52.

48. The adjustment of the length of the vector arm 45 may be taken care of by a clamping screw 55. As illustrated, the adjustable vector arm 48 is provided with a graduated scale 52 to facilitate the adjusting o the length of the arm. Mounted upon the end of the adjustable vector arm 48 is a graduated scale member 58 which may be rotated in a plane parallel to the base 24 about a pivot member 54 which connects the graduated scale member 58 to the end of the adjustable vector arm 48. The point z lies in the center of the pivotI member 54. The circular scale member 58 and the adjustable vector arm 48 may be adjusted angularly with reference to the rotating shaft 42 by means of a set screw which threadably engages the upper end of the rotating shaft42. I

An adjustable vector arm 59, which takes care of the distance between the points :c and y on Figure 2, for example, is pivotally mounted to a swinging arm 51.l A circular scale member 58 is likewise adapted to be rotatably connected to the swinging arml 51. The adjustable vector arm 59 and the circular scale member 58 are arranged to rotate about a shaft of which the point a: is the center and they both may be adjustably set with reference to the shaft about which they rotate by means of a thumb set clamp 5 I, which when loose permits the adjustable vec- -tor arm 59 and the circular scale member 58 to move relative to the shaft about which they are mounted and which when set or tightened secures the adjustable vector arm 58 and the circular scale member 59 to the shaft about which they revolve. The shaft to which the adjustable vector arm 59 and the circular scale member 59 revolve may be indicated by the rei'- erence character 55 and to this shaft is secured a pulley 50 driven bya flexible member such, for example, as a flexible wire indicated by the One end of the wire 52,Y is anchored to an anchoring member 54 after which it passes over a pulley 55, upon the lefthand side thereof, and then across to the pulley 55 where it is wrapped around one or more times and then back across to the right-hand ments of the traverse member 84 are transmitted swinging arm 51 is secured in a fixed adjustable position by means of a hand set clamp 58 which threadably engages the upper end of the stationary post about which the swinging arm 51 revolves. The adjustable swinging of the swinging arm l1 provides for taking care of the distance between the points w and :c such, for example, as shown on Figure 2 of the drawings. The. swinging arm 51 is elevated above the plane of the base 24 so that the adjustable vector arm 58 and the graduated scale member 10 may swing in a plane which' is above the plane in which the adjustable vector arm 45 and the-graduated scale member 5l attached thereto is adapted to rotate. Therefore, the swinging arm 51 may be oscillated in its elevational plane without interfering with the adjustable vector arm 48 and the scale 53 pivotally connected thereto. The distance between the points w and .1: in Figure 7 correspond to the distance between the points w and for example, in Figure 2 wherein the two points w and :c represent the ends of a vector Vi, which may be designated as a reference vector. In other words, the shafts 42 and 56 represent the ends of a reference vector.

The time phase angle pz of the current owing in antenna No. 2 with respect tothe current flowing in antenna No. I is set of! on the circular scale member 55, using the line passing through the points w and :l: as a reference line. The time phase angle if: of the current flowing in antenna No. 3 with respect to the current flowing in antenna No. l is set off on the circuto the pulley 59 for rotating the adjustable vector arm 59 and the circular scale member 58 about 55. As illustrated, the adjustable vector arm 58 is slidably mounted within an elongated block 55 which is adjustably secured to the shaft by the thumb set clamp 5I. Upon the end of the adjustable vector arm 58 is pivotally mounted a graduated scale member 10 whichmay rotate in a plane parallel to the base 24 about a pivot member 55 which connects'the scale member 18 to; the end of the adjustable vector arm 59. 'I'he point y lies in the center of the pivot member 55. The adjustable vector arm 58` is provided with a graduated scale 51 which facilitates the adjusting of the length of the arm between the ypoints a: and y. When the adjustable vector arm 59 is once positioned as to its length, it'may be set by means of the clampin screw 11. 4

The swinging arm 51 is arranged to be swung in a plane parallel to the base 24 about a stationary post mounted upon the upper side ofthe base 24 near the traverse member 34. The

lar scale member 58, using the line passing through w and a: as a reference line. In other words, the zero setting for the scale members 58 and 68" coincide with the reference line passing 'through the points w and As will be observed, the reference line w and :c assumes a different position as the swinging arm 51 is moved to vary the distance between `the points w and I. In my invention the reference line is scribed of! on a separate piece of material or a plate indicated generally by the reference character 12. This plate may be made of relatively thin material and is positioned directly upon the base 24 and is arranged to pivot about the shaft 42. The reference line which is marked upon the plate 12 is indicated by the reference character 1| and extends from one side of the plate to the other side. The plate is shifted to always keep the lreference line 1I in a plane which passes through the `points w and :t on the shafts 42 and 55. The shifting of the plate 12 about the pivot point of the shaft 42 is arranged to be accomplished by a lever arm 15 having a pin 14 carrid thereby which fits within an elongated slot 15. Thelever arm 13 has its other end pivotally connected about the stationary post upon which the swinging arm 51 is rotated. The iev ver arm 13 lies directly on top of the thin plate 12 with the pin 14 engaging the elongated slot 15 and the ,curvature of the slot 15 is such that for any position of the lever arm` 18 lthe reference line 1i always passes through -a plane which contains the points w and .'r.- A scale 18 is po-A sitioned adjacent the elongated slot'15 and is calibrated to represent the distance between the points w and x. In other words, the value of the vector V1 is set oifupon the scale 18. The circular scale member 58 is positioned immediately on top of the plate 1-2 and thus an observer can readily align the/marks of the circular scale member `58 with the vreference line 1t. However, the circular scalemember, since it is mounted upon the swinging arm 51 at an elevation above the plate 12 and the reference line 1I, becomes more diilicult to align the marks thereon with the reference line 1I. To facilitate the alignment o! the marks upon the circular scale member 88 with the reference line 1I spaced therebelow by an amount equal to the distance that the circular scale member 69 is positioned above the plane of the base 24, I provide an alignment element 1B which extends radially outwardly from the circular scale member 69. The alignment element 15 may be provided with a sighting line 19 so that the operator may align the sighting line with the reference line 1 I The left-hand end of the alignment element 15 is` carried by the swinging arm 51 and is arranged to be pivotally connected thereto so that it may be irictionally moved to align the sighting line 19 thereon with the reference line II on the plate 12, After the alignment element 1B is positioned so that `the sighting line 18 lies in a vertical plane perpendicularly above the referencef line 1I, then the mark on the circular scale member G9 may be made directly with reference to the sighting line 19 instead of with reference to the line 'II positioned at a spaced distance' therebelowz In other words, the alignment element 15 and the sighting line 19 thereon serve as means to facilitate the setting oi' the circular scale member with reference to the line 1I on the plate 12.

,The resultant of the vector is the distance between the points s and y and represents a value which is a. function of the relative ii'eld intensity in any direction, both horizontally and vertically. The distance between the points z and ymay be directly read bythe two scales 3 and 10 pivotally connected to theA ends of the adjustable vector arms 4l and 5l. The scales 53 and 1l both carry marks thereon which when added together give the distance between the points z and y. inasmuch as the scales 53 and 10 are pivotally connected to the ends of the adjustable vector arms II and 59, they may be aligned with each other in any direction to read the distance be- The iield V1k from the antenna No. I is set oil on the scale 18 so that the pointerof the arm 13 coincides with themark 2.5. The ileld V2 from the antenna, No.1. yis set oiI on the adjustable vector arm-'5.9 so that the graduated scale 81 reads 1.4. The field V: from the antenna No. 3 is set oi on the graduated scale 52 on the adjustable vector arm Il at a value of 1.3. The spacing t Se between antenna No. I and antenna No. .i

is set oi! on the graduated scale 28 of the adJusting arm ZD-at a value of 200. 'I'hespacing Sa between the antenna No. I and the antenna No. I is set on on the graduated scale of the adjusting arm Non the underneath side oi the base asesora plate 24 at a value oi.' 180. The angle on is set ofi by angularly positioning the antenna No. 2 at 210K degrees measured upon the inner circular scale for degrees. The angle qb: is set of! by angularly positioning the antenna A: on the under- 'neath side of the plate 24 at 330 degrees upon the inner circular scale for degrees. The phase angle ill: is set ofi' on the circular scale member E! at a value of 220 degrees measured with reference to the reference line 1I. The phase angle pa is set of! on the circular scale member Bl at a value of degrees with reference to the reference line 1I or the sighting line 19 on the alignment element 1I.

The setting of the cosine generator l5 is shown diagramm'atically in Figure 1 and the setting oi' the vector system I8 is shown diagrammatically in Figure 2. Inasmuch as the magnitude of the eiective fields radiated by the antenna elements Nos. I, 2 and 3 are respectively proportional to the current owing in the respective antenna elements, I illustrate the magnitude of the vector from w to :c as V1, the magnitude of the vector from to y as Vn, the magnitude or the vector w to z as V3, and the magnitude oi' the resultant vector from z to y as V which represents the value of the radical in Equation No. 4. It is to be observed with reference to Figure 2 that the angle between the vector V1 and the vector VJ is measured or indicated by ai, which in turn equals Sa cos (fio-ip) +903. Therefore, in settin oil the angle on the circular scale member 5I with reference to the reference line 1I, the expression S3 cos (ea-4:) is iixed at zero and this is done by rotating the .indicator pointer I until theantenna A: is vertical after which the angle of ipa=140 degrees may be' directly set ci! on the circular scale' member B8. In a similar manner the antenna Az is moved to a vertical position so that the expression Sn cos (z-) equals zero when setting *lf2 oil! on the circular scale member 59 with reference to the reference line 1I or the sighting line 19 of the alignment element 15.

When the machine is once setoi, the magnitude oi' ,the resultant vector V, being the distance from z to y, is measured directly by counting the sum of the marks on the scales 53 and 1| between the points z and y. In Figure 7 the pointer I is directed to the zero degree reading on the outer circular scale which means that the angle qb iszero degrees. The value ol.' the magnitude ci' the resultant vector V for the setting as shown in Figure 7, is 4.04. The next reading of the resultant magnitude of the vector V is taken whenA the pointer I is placed upon the 10 degree mark for the outer circular scale of the cosine generator. `The readings for the magnitude of the resultant vector V are taken at intervals of 10 degrees for e until the complete cycle of one revolution is nished. For an angle oi =70 degrees, the antennas A: and La of Figure 'I' assume a position as shown diagrammatically in Figure 3 and the system of vectors will assume a position as diagrammatically shown in Figure 4. I'he individual values oi. the magnitude oi the resultant vector V as determined by the readings on the scales 53 and 10 of my machine in Figure 7 are ultiplied by a constant K for obtaining the relative iield strength being radiated. The plotting of the values for, e,- at 10 degree intervals for the complete cycle of 360 degrees gives the Thorizontal pattern, such for example as illustrated in Figure 5. -The values for, e, at p equals zero degrees and for, e, at 0 equals 70 degrees is shown by the polar vector lines in Figure 5.

calculated by means of a slide rule which takes care of the factorsroutside of the radical in Equation 2. The various vertical plane patterns represent conditions in direction at various.

angles above the horizon, in which the angles may be designated by the angle 0. In' solving for a value of the eld intensity at P, or a vertical pattern, the location of P is determined by two angles, namely, and 0. The relative position of the observation point P with reference to the three antennas for the vertical pattern is shown by the combined illustrations in Figures 8 and 9, wherein' equals 20 degrees as shown in Figure 8 and 0 equals 60 degrees as shown in Figure 9. The angle is measured from the verticalbecause 0 equals the zenith angle where the vertical is; considered zero degrees.

In solving for the value of the radical is Equation 2 for obtaining a value of the field intensity for the antenna array at the observation point P shown, for example, vin Figures 8 and 9, my indicator I is first moved to 20 degrees on the outer circular scale for degrees. In this position the antennas Anand A3 will assume a position as indicated diagrammatically in Figure 8. This setting takesl care of the factor, =20 degrees. 'I'he next operation in setting up my machine to obtain values for determining the vertical patterns of an antenna array'is to lock the traverse mem- 'bers M and 31 in a fixed position. s As illustrated,

this may be accomplished by means of hand operated cams 80 and 8l which when turned engage the sides, respectively, of the vtraverse members 34 and 31. The cam 8|! is mounted above the main base plate 24 and the cam 8| is mounted beneath the base plate 1,34*. When'the cams 80 and 8i are clamped respectively against `the traverse members 34 and 31 they a're constrained ina fixed position as determined by the setting of the pointer I on the 20 degree mark on the outer circular scale for degrees. for setting up my device to measure values for the vertical pattern is to loosen the adjusting arms 20 and 29 of the cosine generator and position them in horizontal alignment, the antenna A3 pointing lto the mark zero on the outer circular` zero scale and the antenna A: pointing tow'ard the mark 180 degrees upon .the outer Y circular degree scale. `The pointer I is lalso -returned to the zero mark on the outer circular de.

gree scale. The position of the antennas A1. Aa and A3 and the indicator I for the setting just described is the same as that illustrated diagrammatical'y in Figure 9. The antennas An and As and the adjusting arms 20 and 29 which carry them respectively are retightened, and the hand operated cams at and 8| are operated to Yfree the traverse members 3l and 31. The machine is now set to take readings at intervals of 10 degrees for the pointer I from horizontal to 90 degrees. the vectors in my vector system IS will be the the scales 53 and 10 are tabulated or recorded at The next operation Iii the 10 degree intervals for the pointer AI between the horizontal and 90 degrees. The position of the vectors in my vector system I! will be the same as that shown diagrammatically in Figure 10 for an angle 0 equals 60 degrees. The values as indicated upon the scales I3 and III for the resultant vector V. which represents the value of the radical in the Equation 2, are then multiplied by the coeiiicient of the radical to obtain the relative eld intensity, e, for values for the vertical pattern. 'I'hese values may then be plotted as shown in Figure 11 from the horizontal to 90 degrees. The line OP represents the relative field intensity for equals 20 degrees and 0 equals degrees. In plotting Figure 11, the

`antennas were assumed to be 90 degrees in height.

In determining the vertical values with my machine, the pointer I is moved at 10 degree intervals with reference to the outside circular degree scale from the horizontal to 90 degrees, but the value for 0 is a measure of the angle between the indicator I and the lvertical of 90 degrees. In

other words, for example, when the pointer I is 4 set upon the 30 degree markfthe actual value for 6 is 60 degrees because 0 is the zenith angle where the vertical is considered zero degrees.

The Figure 12, when taken in combination with the Figure 3 is another diagrammatic representation for equals '70 degrees and 0 equals 40 degrees. The Figure 13 yis a representation of the position which my vector system I6 will assume for equals 70 degrees and 6 equals 40 degrees. The'Figure 14 is a vertical pattern at equals degrees whenall the antennasare 90 degrees in height and for 0 taken from the Ahorizontal to degrees. The line OP in Figure 14 represents the relative held intensity at the angle 0 'equals 40 degrees as illustrated in Figure l2.

It is to be noted that my machine functions to solve for the value of the radical in the Equa- `calculated, by multiplying the value ot the resultant vector by the coeillcient of the radical.

My machine is directly applicable to solve values i'or a two-element antenna array. This may be done by positioning the point z at the end of the, adjustable vector arm l! directly over the point w of the rotating shaft 42,l which is referred to in some of the claims as a member having thereony a reference point. This means that the antenna As becomes ineffective regardless of its positional setting in moving the point 2, since it is placed directly above the point w. The relationship between the antenna A1 and Az remains operative in my machine to solve for values for the two-elementl antenna array. n In a broad sense, my mechanism is adapted to solve the equation which may be represented by the general vector equation,

In a broad structural sense, my invention comprises a mechanism for generating a movement corresponding to a cosine function and for transferring the said movement to a plurality of conditions which when affected by the generated movement provides for giving a resultant condition at any sition throughout the complete cycle oi' the generated movement.

I vclaim as my invention:

1. A machine for solving equations which may be solved vectorially comprising, in combination, a movable element having thereon two spaced points constituting the ends of a movable vector, a member having thereon'a reference point, said reference point and one of said spaced 'points of the movable vector being spaced apart and constituting the ends of a reference vector, the said reference point and the other of said spaced points of the movable vector constituting the ends of a resultant vector representing a function of the value to be solved, said movable element being rotatable about the said one of said spaced points thereon, means for determining the lresultantvector representing a function of the value to be solved, means for generating a recurrent movement, and means for transmitting the generated movement to the movable element to rotate same about the said one of said spaced points thereon for giving the resultant vector at any position throughout the complete cycle of the recurrent movement.

2. A machine for solving equations which may be solved vectorially comprising, in combination, a movable element having thereon two spaced points constituting the ends of a movable vector,

reference point and one of said spaced points of the movable vector being spaced apart and constituting the ends of a reference vector, the said reference point and the other of4 said spaced points of the movable vector constituting the ends oi' a resultant vector representing a function of the value to be solved, said movable element being rotatable about the said one of said spaced points thereon,X means for determining the resultant vector representing a function oi the value to be solved, means Afor generating a recurrent movement, means for transmitting the generated movement to the movable element to rotate' same about the'said one of said 'spaced points thereon for giving the resultant vector/ at any position throughout the complete cycle of the recurrent movement, 'and manually adjustable means'for varying the distance between the .35 be solved vectorially comprising, in combination, a first movable element having thereon two spaced two spaced points on the movable element.

3. A machine for solving equationswhich may be solved vectorially comprising, in combination, a movable element having thereon two spaced points constituting the ends of a movable vector,

5 complete cycle of the recurrent movement, manually adjustable means for varying the distance between the two spaced points on the movable element, and manually adjustable means for varying the distance between the reference point on the said member and said one of said spaced points of the movable vector.

5. A machine for solving equations which may be solved vectorially comprising, in combination, a movable element having thereon two spaced points constituting the ends of a movable veca member having thereon a reference point, said tor, a member having thereon a reference point, said reference point and one of said spaced points of the movable vector being spaced apart and constituting the ends of a reference vector, the

said reference point and the other of said spaced points of the movable vector constituting ends of-"a resultant-'vector representing: a func-- tion of the value to be' solved, said-movable element being rotatable about the said one of said spaced points' thereon, meansfor determining to rotate same labout thel said vone-of said spaced points thereon for giving'the resultant vector at any position throughout the'complete cycle of the cosinefmovement: Y

6. A machine for solving equations which may points constituting theends of a ilrst movable vector, a secondmovable element having there- @on two spaced points constituting the ends oi a member having thereon a reference point, said A0" a 'second movable vector, one oi said spaced points reference point and one of said spaced points of the movable vector being spaced apart and constituting the ends of a reference vector, the said reference point and the other of said spaced on the first movable vector and one oi said spaced points on the second movable vector being spaced apart and constituting the ends of a reference vector, the other of said spaced points on the points of the movable vector constituting the rst movable vector and the other of said spaced ends of a resultant vector representing a iunction of the value to be solved, said movable element being rotatable about the said one of said spaced points thereon, means for determining the resultant vector representing a function of the value to be solved, means for generating a Vrecurrent movement, and means including translatory means and rotary means for transmitting the generated movement to the movable elementl to rotate same about the said one of said spaced points thereon for giving the resultant vector at any position throughout the complete cycle of the recurrent movement.

4. A machine for solving equations which may be solved vectorially comprising, ingcombination, a movable element having thereon two spaced points constituting the ends ci a movable vector, a member having thereon a reference point, said reference point and one of said spaced points of the movable vector being spaced apart and constituting the ends of a reference vector, the said reference point and the other of said spaced points of the movable vector -constituting the ends of a resultant vector representing a function of the value to be solved, said movable element being rotatable about the said one of said spaced points thereon, means ior` determining the resultant vector representing a function of the value to be solved, means for generating a recurrent movement, and means for transmit- 75 points on the second movable vector constituting the ends of a resultant vector representing. a function of the value to be solved, said first movable element being rotatable about the said 'one of said spaced points thereon, said second movable element being rotatable about the said one of said spaced points thereon, means for determining the resultant vector representing a function of the value to be solved, means for generating a recurrent movement, and means ior transmitting;` the generated movement to the first andsecond movable elements torotate same respectively about the said one of said spaced points thereon for giving the resultant vector at so any position throughout the complete cycle of the recurrent movement.

'7. A machine for solving equations which may be solved vectorially comprising, in combination, a ilrst movable element having thereon two spaced points constituting the ends of a first movable movable vector, manually adjustable means ior varying the distance between said two spaced points one of said spaced points on the ilrst mov- -able vector and one of said spaced points on the second movable vector being spaced apart and constituting the ends of a reference vector, the

other of said spaced points on the first movable vector and the other of said spaced points on the second movable vector constituting the ends of a resultant vector representing a function of the value to be solved, said first movable element being rotatable about the said one of said spaced points thereon, said second movable element being rotatable about the said one of said spaced points thereon, means for determining the resultant vector representing a function of the value to be solved, means for generating a' re current movement, and means for transmitting the generated movement to the first and second movable elements to rotate same respectively about the said one of said spaced points thereon for giving the resultant vector atany position throughout the complete cycle of the recurrent movement.

8. A machine for solving equations which may be solved vectorially comprising, in combination, a first movable element having thereon two spaced points constituting the ends of a first movable vector, a second movable element having thereon two spaced points constituting the ends of a second movable vector, one of-said spaced points on the first movable vector and one of said spaced points on the second movable vector being spaced apart and constituting the ends of a reference vector, the other of said spaced points on the first movable vector and the other of said spaced points on the second movable vector constituting the ends of a resultant vector representing a function of the value to be solved, said first movable element being rotatable about the said one of said spaced points thereon, said second movable element being rotatable about the said one of said spaced points thereon, means for determining theqresultant vector representing a function of the value to be solved, means for generating a recurrent movement, and means including translatory means and rotary means for transmitting the generated movement to the iirst and second movable elements to rotate same respectively about the said one of said spaced points thereon for giving the resultant vector at any position throughout the complete cycle of the recurrent movement.

9. A machine for solving equations which may rst movable vector and the other `of said spaced be solved vectorially comprising, in combination,

a rst movable element having thereon two spaced points constituting the ends of a first movable vector, a second movable element having thereon two spaced points constituting the ends of a second movable vector, one of said spaced points on the first movable vector and one of said spaced points on the second movable vector being spaced apart and constituting the ends of a reference vector, vthe other of said spaced points on the first movable vector and the other of said spaced points on the second movable vector constituting the ends of a resultant vector representing a function of the value to be solved, said rst movable element being rotatable about the said one of said spaced points thereon, said 'second movable element being rotatable. about the said one of said spaced points thereon, means for determining the resultant vector representing a function of the` value to be solved, means for generating a recurrent movement, and means for transmitting the generated movement to the first and second movable elements to rotate same respectively about the said one of said spaced points 'thereon for giving the resultant vector at any position throughout the complete cycle of the recurrent movement, said generating means comprising a rotating arm and a translatory member interconnecting the rotating arm and the transmitting means to impart a cosine movement to the transmitting means.

10. A machine for solving equations which may be solved vectorially comprising, in combination, a first movable element having thereon two spaced points constituting the ends of a first movable vector, a second movable element having thereon two spaced points constituting the ends of a second movable vector, one of said spaced points on the first movable vector and one of said spaced points on the second movable vector being spaced apart and constituting the ends of a lreference vector, the other of said spaced points on the points on the second movable vector constituting the ends of a resultant vector representing a function of the value to be solved, said flrst movable element being rotatable about the said one of said spaced points thereon, said second movable element being rotatable about the said one of said spaced points thereon, means for determining the resultant vector representing a function of the value to be solved, first cosine means for generating a first movement corresponding to a cosine function, second cosine means for generating a second movement corresponding to a cosine function, means for transmitting the first generated movement to the first movablev element to rotate same about the said one of said spaced points thereon and means for transmitting the second generated movement to the second movable element to rotate same about the said one of said spaced points thereon, the rotation of the first and second movable element giving the resultant vector at any position throughout the complete cycle of the recurrent movement.

11. A machine for solving equations which may be solved vectorially comprising, in combination, a first movable element having thereon two spaced points constituting theends of a first movable vector, means for varying the distance between said two spaced pointsga second movable element having thereon two spaced points constituting the ends of a second movable vector, means for varying the distance between said two spaced points, one of said spaced points on the first movable vector and one of said spaced points on the second movable vector being spaced apart and constituting the ends of a. reference vector, means for varying the length of thefreference vector, the other of said spaced points on the first movable vector and the other of said spacedl points on the second movable vector constituting the ends of a resultant vector representing a function-of the value to be solved,said first movable element being rotatable about the said one of said spaced points thereon, said second movable element being rotatable about the said one of said spaced points thereon, means for determining the resultant vector representing a function of the value to be solved, means for generating a recurrent movement, and means for transmitting the generated movement to the first and second movable .elements to rotate same respectively about the said one of said spaced points thereon for giving the resultant vector at any position throughout the complete cycle of the recurrent combination, a plurality of means adapted to give a resultant condition representing -the value of e, said plurality of means comprising at least a first means establishing a condition representing the. value of Kili and a second means establishing a condition representing the magnitude of KV, means for generating a movement corresponding to a cosine function, means for transmitting the generated movement to the second means to modify the second means for establishing a condition representing the value of Kiln, and means governed by the first and second means for giving the value of e at any position throughout the complete cycle of the generated movement.

13. A machine for solving an equation of the general type, e=K(ll1-{- fn) comprising, in

combination, a plurality of means adapted to give a resultant condition representing the value of e, said plurality of means comprising at least a first means establishing a condition representing the value of Kil/1 and a second means establishing a condition representing the magnitude of KV, said second means including a rotating element, means for generating a movement correspending to a cosine function, means for transmitting the generated movement to the rotating element to modify the second means for estab- `lishing a condition representing the value of Kiln.

and means governed'by the first and second means for giving the value of e at any position throughout the complete cycle of the generated movement.

14. A Vmachine for solving an equation of the general type, e=K (V1-+- ilu) comprising, in

combination, a plurality of means adapted to give a resultant condition representing the value of e, said plurality of means comprising at least a first means establishing a condition representing the value of KV1 and a second .means including a rotating element and vector member for establishing a condition representing lthe magnitude of KVn, means for generating a move-- ment corresponding to a cosine function, means for transmitting the generated movement to the second means to modify the second means for establishing a condition representing the value Kiln, meansfo'r generating a recurrent movement, means for transmitting the generated movement to the second means to modify the secf ond means for establishing a condition representing the value of Kiln, and means governed by the first and second means for givingthe value of e at any position throughout the complete.

cycle of the generated movement.

16. A machine for solving an equation of the general type, e=K(V1-|- Vn) comprising, in in combination, a plurality of means adapted to give a resultant condition representing the value of e, said plurality of means comprising at least a first means establishing a condition representing the value of Kili, a second means establishing a condition representing the magnitude ol KVz, a third means establishing a condition representing the magnitude of KVn, means for generating a recurrent movement, means for transmitting the generated movement to the second means to modify the second means for establishing a condition representing the value of KVz, means for transmitting thegenerated movement to the third means to modify the third means for establishing a condition representing the value of KVn, and means governed by the'rst, second and third means for giving the value of e at any position throughout the complete cycle of the generated movement.

'17. A machine for solving an equation of the .general type,e=K(ll1-l-i`l2-{- ilu) comprising,

in combination, a plurality of means adapted to give a resultant condition representing thevalue of e, said plurality of means comprising at least a first means establishing a condition representing the value of KVi, asecond means establishing` a condition representing the magnitude of KV2, a third means establishing a condition representing the magnitude of KV", first cosine means for generating a first movement corresponding to a cosine function, second cosine means for generating a second movement corresponding to a cosine function, means for transmitting the first generated movement to the second means to modify the second means for establishing a condition representing the value oi Kilz, means for transmitting the second generated movement to the third means to modify the third means for establishing a condition representing the value of KVn, and means governed by the first, second and third means forgiving the value of e at any position throughout the complete cycle of the cosine generated movements.

18. A machine for solving ain-equation of the general type, e=K(V1-}-V2-{ Vn) comprising, in combination, a plurality of means adapted to give a resultant condition representing the value of e, said plurality of means comprising at least a first means establishing a condition representing the value of KV1, a second means establishing a condition representing the magnitude of KV2, a third means establishing a condition representing the magnitude of KV, first cosine means for generating a first movement corresponding to a cosine function, second cosine means for generating a second movement corresponding to a cosine function, means for transmitting the first generated movement to the second means to modify the second means for establishing a condition representing the value of Kllz, means for transmitting the second gen` erated movement to the third means to modify the third means for establishing a condition representing the value-of KV". and means governed by the first, second and third means for giving the value of e at any position throughout the complete cycle of the cosine generated movements, and means for adjustably shifting one of said cosine generating means relative to the other.

19. A machine for solving an equation of the general type, e=K(l/"1|- Vn) comprising, in Combination. a plurality of means adapted to give a resultant condition representing the value of e, said plurality of means comprising at least a first means establishing a condition represent-4 tion representing the value of KVn, and means governed by the rst and second means for giv ing the value of e at any actuated position of the third means.

20. A machine for producing a resultant condition from a'plurality of individual conditions comprising, in combination, a plurality of means adapted to give the resultant condition, said plurality of means comprising at least a rst means establishing a rst individual condition, second means including a rotating element for establishing a secondI individual condition, means for generating a movement corresponding to a cosine function, means for transmitting the generated cosine movement to the rotating element to modify the condition of the second means, and means governed by the first and second means for giving the said resultant condition.

WILLIAM G. HUTTON. 

